在Liao（2018）和Peng（2018）中，提出本質編碼與本質效應的觀念。其所討論的數據型態是反應變數為函數型變數，而解釋變數為純量型變數。模型則為$Y(t) = \beta_0 (t) + \bm X \bm \beta (t) + \epsilon(t)$，並假設$\bm \beta(t) = \bm \Gamma \bm \Phi(t)$，亦即函數型效應$\bm \beta (t)$為由數個未知的平滑本質編碼$\bm \Phi(t)$，經本質效應$\bm \Gamma$的線性轉換後所形成。雖然$\bm \Phi(t)$假設為平滑函數，但在Liao（2018）和Peng（2018）中，在估計$\bm \Phi(t)$時，都沒有考慮$\hat{\bm \Phi}(t)$的平滑化問題。本文主要探討在考慮平滑性的觀點下，如何估計平滑本質編碼。我們使用的方法為正規化法，其在原$\bm \Phi(t)$的最佳化準則中加入量化$\bm \Phi(t)$之粗糙性的懲罰項，並以$\bm \Phi(t)$的二階導函數來量化其粗糙程度，但因$\bm \Phi(t)$可為任意二階可微之未知函數，故無法明確地求得其二階導函數。為了解決此問題，我們使用$\bm \Phi(t)$的二階差分來近似之。故本文結合正規化法與不同的差分近似手法，以估計出重要的平滑本質編碼。在正規化法中，必須給定一個平滑參數來控制懲罰項的比重，以在最佳化準則和$\hat{\bm \Phi}(t)$的平滑程度之間折衷。本文除了考慮主觀給定的平滑參數外，亦結合交叉驗證法的精神，分別探討在所有本質編碼使用相同平滑參數和不同本質編碼各自使用不同平滑參數下，發展出利用數據客觀地選出合適的平滑參數之方法。最後，本文將上述方法應用於晶圓資料上。並比較所有平滑本質編碼對應相同或相異平滑參數、各種估計準則、各種平滑參數設定方法、各種交叉驗證選取平滑參數之方法、各種差分近似方法在不同晶圓模擬資料上，其所估計之平滑本質編碼的異同。

Liao (2018) and Peng (2018) proposed the concept of essence codings and essence effects for functional linear models, in which the response is a function and the explanatory variables are scalar. They considered the model $Y(t) = \beta_0 (t) + \bm X \bm \beta (t) + \epsilon(t)$, and assumed that the coefficient function $\bm \beta(t)$ is of the form $\bm \beta(t) = \bm \Gamma \bm \Phi(t)$, where $\bm \Phi(t)$ is the vector of essence codings and $\bm \Gamma$ is the matrix of essence effects. Although every essence codings in $\bm\Phi(t)$ are assumed to be a smooth function, Liao (2018) and Peng (2018) did not include a smoothing step in their estimation procedure of ${\bm\Phi}(t)$. In this thesis, we adopt the regularization-based technique to improve their estimation and create smooth solutions by adding a penalty term to their estimation criteria. The penalty term quantifies the roughness (or smoothness) of essence codings as the integrated square of the second derivative of ${\bm\Phi}(t)$. Since the essence codings in $\bm \Phi(t)$ are unknown functions, we approximate their second derivatives by the method of (second-order) finite difference. The degree of regularization is controlled by a smoothing parameter, which balances the trade-off between original estimation criteria and smoothing. In our method, the smoothing parameters can be identical or different for every essence codings. Our method also allow their values to be subjectively assigned by users or objectively determined using cross-validation. The method developed in this thesis is applied on several simulated functional data of wafer thickness to compare the difference in the estimated essence codings between various estimation criteria, various settings of smoothing parameters, and various finite-difference approximations.

1 緒論1
2 文獻回顧7
2.1 有限差分運算. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 本質編碼之估計準則. . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 研究方法14
3.1 特徵分解法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 正交空間法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 平滑本質編碼之估計準則. . . . . . . . . . . . . . . . . . . . . . . 17
3.4 交叉驗證法選取平滑參數. . . . . . . . . . . . . . . . . . . . . . . 18
3.4.1 相同λ法. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4.2 相異λ法. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 數據分析23
4.1 晶圓厚度模擬資料. . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 平滑本質編碼之估計結果. . . . . . . . . . . . . . . . . . . . . . . 24
4.3 結果比較. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3.1 相同和相異平滑參數的結果比較. . . . . . . . . . . . . . . 28
4.3.2 三個估計準則的結果比較. . . . . . . . . . . . . . . . . . . 30
4.3.3 主觀和客觀選取平滑參數的結果比較. . . . . . . . . . . . . 30
4.3.4 使用S1和S2差分近似的結果比較. . . . . . . . . . . . . . 31
4.3.5 不同Σ之晶圓模擬資料的結果比較. . . . . . . . . . . . . 32
4.3.6 模擬資料有和無重複曲線之結果比較. . . . . . . . . . . . . 32
5 結論與建議34
參考文獻35
附錄36

[1] Liao, Y.-F. (2018). Identifying essence codings and effects in functional linear models with homogeneous and independent errors, Master thesis, National Tsing Hua University, Hsinchu, Taiwan.
[2] Okada, T. and Tomita, S. (1985). An optimal orthonormal system for discriminant analysis. Pattern Recognition, 18(2), 139-144.
[3] Peng, P.-R. (2018). Identifying essence codings and effects in functional linear models with heterogeneous and correlated errors, Master thesis, National Tsing Hua University, Hsinchu, Taiwan.
[4] Ramsay, J.O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd edition, Springer, New York.
[5] Yang, W. Y., Cao, W., Chung, T. S, and Morris, J. (2005). Applied Numerical Methods Using MATLAB. John Wiley & Sons, Hoboken, New Jersey.